Namespaces
Variants
Actions

Difference between revisions of "Zhegalkin algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
The special algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992701.png" />, where
+
<!--
 +
z0992701.png
 +
$#A+1 = 16 n = 0
 +
$#C+1 = 16 : ~/encyclopedia/old_files/data/Z099/Z.0909270 Zhegalkin algebra
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992702.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992703.png" /> is the multiplication operation. The [[Clone|clone]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992704.png" /> of the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992705.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992706.png" /> is of interest. Every operation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992707.png" /> is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992708.png" />, a so-called Zhegalkin polynomial, named after I.I. Zhegalkin, who initiated the investigation of this clone [[#References|[1]]]. He proved that every finitary operation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z0992709.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927010.png" />. Thus, the study of properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927011.png" /> includes, in particular, the study of all algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927012.png" /> for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927013.png" />.
+
The special algebra  $  \mathfrak A = \langle  A , \Omega \rangle $,
 +
where
 +
 
 +
$$
 +
= \{ 0 , 1 \} ,\ \
 +
\Omega  = \{ {x \cdot y } : {x + y \
 +
(  \mathop{\rm mod}  2 ) , 0 , 1 } \}
 +
,
 +
$$
 +
 
 +
and  $  x \cdot y $
 +
is the multiplication operation. The [[Clone|clone]] $  F $
 +
of the action of $  \Omega $
 +
on $  A $
 +
is of interest. Every operation in $  F $
 +
is a polynomial $  \mathop{\rm mod}  2 $,  
 +
a so-called Zhegalkin polynomial, named after I.I. Zhegalkin, who initiated the investigation of this clone [[#References|[1]]]. He proved that every finitary operation on $  A $
 +
is contained in $  F $.  
 +
Thus, the study of properties of $  F $
 +
includes, in particular, the study of all algebras $  \mathfrak A = \langle  A , \Omega  ^  \prime  \rangle $
 +
for arbitrary $  \Omega  ^  \prime  $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Zhegalkin,  ''Mat. Sb.'' , '''34''' :  1  (1927)  pp. 9–28</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.V. Yablonskii,  G.P. Gavrilov,  V.B. Kudryavtsev,  "Functions of the algebra of logic and Post classes" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Zhegalkin,  ''Mat. Sb.'' , '''34''' :  1  (1927)  pp. 9–28</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1986)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.V. Yablonskii,  G.P. Gavrilov,  V.B. Kudryavtsev,  "Functions of the algebra of logic and Post classes" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In other words, the Zhegalkin algebra is the two-element [[Boolean ring|Boolean ring]], the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927014.png" /> or the free Boolean algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927015.png" /> generators. As such, it is generally not given a distinctive name in the Western literature. Cf. e.g. [[Boolean algebra|Boolean algebra]]; [[Boolean equation|Boolean equation]]. The study of all algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099270/z09927016.png" /> is the subject of E.L. Post's dissertation [[#References|[a1]]].
+
In other words, the Zhegalkin algebra is the two-element [[Boolean ring|Boolean ring]], the field $  \mathbf Z /( 2) $
 +
or the free Boolean algebra on 0 $
 +
generators. As such, it is generally not given a distinctive name in the Western literature. Cf. e.g. [[Boolean algebra|Boolean algebra]]; [[Boolean equation|Boolean equation]]. The study of all algebras $  \mathfrak A = \langle  A, \Omega  ^  \prime  \rangle $
 +
is the subject of E.L. Post's dissertation [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Post,  "Two-valued iterative systems of mathematical logic" , Princeton Univ. Press  (1941)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Post,  "Two-valued iterative systems of mathematical logic" , Princeton Univ. Press  (1941)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


The special algebra $ \mathfrak A = \langle A , \Omega \rangle $, where

$$ A = \{ 0 , 1 \} ,\ \ \Omega = \{ {x \cdot y } : {x + y \ ( \mathop{\rm mod} 2 ) , 0 , 1 } \} , $$

and $ x \cdot y $ is the multiplication operation. The clone $ F $ of the action of $ \Omega $ on $ A $ is of interest. Every operation in $ F $ is a polynomial $ \mathop{\rm mod} 2 $, a so-called Zhegalkin polynomial, named after I.I. Zhegalkin, who initiated the investigation of this clone [1]. He proved that every finitary operation on $ A $ is contained in $ F $. Thus, the study of properties of $ F $ includes, in particular, the study of all algebras $ \mathfrak A = \langle A , \Omega ^ \prime \rangle $ for arbitrary $ \Omega ^ \prime $.

References

[1] I.I. Zhegalkin, Mat. Sb. , 34 : 1 (1927) pp. 9–28
[2] P.M. Cohn, "Universal algebra" , Reidel (1986)
[3] S.V. Yablonskii, G.P. Gavrilov, V.B. Kudryavtsev, "Functions of the algebra of logic and Post classes" , Moscow (1966) (In Russian)

Comments

In other words, the Zhegalkin algebra is the two-element Boolean ring, the field $ \mathbf Z /( 2) $ or the free Boolean algebra on $ 0 $ generators. As such, it is generally not given a distinctive name in the Western literature. Cf. e.g. Boolean algebra; Boolean equation. The study of all algebras $ \mathfrak A = \langle A, \Omega ^ \prime \rangle $ is the subject of E.L. Post's dissertation [a1].

References

[a1] E.L. Post, "Two-valued iterative systems of mathematical logic" , Princeton Univ. Press (1941)
How to Cite This Entry:
Zhegalkin algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zhegalkin_algebra&oldid=49249
This article was adapted from an original article by V.B. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article